Theorem infcntss 8275

Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)

Hypothesis

Ref	Expression
infcntss.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
infcntss	⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Distinct variable group:   𝑥,𝐴

Proof of Theorem infcntss

Step	Hyp	Ref	Expression
1		infcntss.1	. . 3 ⊢ 𝐴 ∈ V
2	1	domen 8010	. 2 ⊢ (ω ≼ 𝐴 ↔ ∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
3		ensym 8046	. . . . 5 ⊢ (ω ≈ 𝑥 → 𝑥 ≈ ω)
4	3	anim2i 592	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ ω ≈ 𝑥) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
5	4	ancoms 468	. . 3 ⊢ ((ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
6	5	eximi 1802	. 2 ⊢ (∃𝑥(ω ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
7	2, 6	sylbi 207	1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1744   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685  ωcom 7107   ≈ cen 7994   ≼ cdom 7995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-er 7787  df-en 7998  df-dom 7999

This theorem is referenced by: (None)